Sources are as follows:
1) He met Yosef whilst he was looking for his brothers and directed him, only for him to then get sold into slavery:
It says in Bereishis 37:15:
וַיִּמְצָאֵ֣הוּ אִ֔ישׁ וְהִנֵּ֥ה תֹעֶ֖ה בַּשָּׂדֶ֑ה וַיִּשְׁאָלֵ֧הוּ הָאִ֛ישׁ לֵאמֹ֖ר מַה־תְּבַקֵּֽשׁ׃
And a man found him, and, behold, he was straying in the field. And the man asked him: What are you seeking?
Rashi identifies this man as the angel Gavriel:
וימצאהו איש - AND A MAN FOUND HIM: This was the angel Gabriel (Genesis Rabbah 84:14) as it is said, (Daniel 9:21) and the man (והאיש) Gabriel”.
The brothers saw him from afar when they threw Yosef into the pit (Midrash Tanchuma, Vayeshev 2:3)
2) Gavriel taught Yosef all 70 languages
Refer to Sotah 36b:
They said to him: If that is so and he is a child of royalty, he should know the seventy languages that all kings’ children learn. The angel Gabriel then came and taught him the seventy languages, but he could not learn all of them. Gabriel then added one letter, the letter heh, to Joseph’s name from the name of the Holy One, Blessed be He, and then he was able to learn the languages, as it is stated: “He appointed it in Joseph [Yehosef] for a testimony, when he went forth against the land of Egypt, the speech of one that I did not know I heard” (Psalms 81:6). And the next day, when he appeared before Pharaoh, in every language that Pharaoh spoke with him, he answered him (sefaria translation and notation)
3) When Potiphar the officer of Pharoah acquired him as a slave he sought to be with him intimately and Gavriel prevented this by castrating him.
It says in Sotah 13b:
The continuation of that verse states: “And Potiphar, an officer [seris] of Pharaoh’s, the captain of the guard, an Egyptian, bought him from the hand of the Ishmaelites, who had brought him down there” (Genesis 39:1). Rav says: He purchased the handsome Joseph for himself, for the intended purpose of homosexual intercourse, but was unable to fulfill his desires, as the angel Gabriel came and castrated Potiphar [seireso]. Then Gabriel came again and further mutilated him [ fero] in the same part of his body. This is alluded to in the verses that write Potiphar’s name differently: Initially, it is written “Potiphar” (Genesis 39:1) and in the end it is written “Poti-phera” (Genesis 41:45). The change in his name indicates that a part of himself was mutilated. (Sefaria translation and notation)
I heard from my Rebbi the following:
The Gemorah in Berachos 4b says that the malach Michoel travels with one wing, whereas Gavriel travels around with two wings (refer to Rashi). The Maharshah explains that Michoel is the malach of Rachamim (mercy) which we want as quickly as possible. However, Gavriel is the malach of Din (judgement) which we want to delay. With this idea we can explain why Gavriel found Yosef looking for his brothers. Targum Yonason Ben Uziel points out that that incident began the enslavement of the Jewish people in Egypt. Since that was a time of מדת הדין, it had to be Gavriel that led Yosef to his brothers, and to eventually be sold to Egypt.
When Yosef first met his brothers after twenty-two years, what would have happened if he would have immediately greeted them and identified himself? Rashi in Parshas Vayechi says that after Yaakov’s death, the brothers thought Yosef would kill them. Yosef calmed them down, saying that if he killed them, people would say they weren’t really his brothers. But Yosef needed people to know they were his brothers to prove he comes from a noble family and not from a slave-family.
With his Rashi, we can explain that if Yosef would have immediately greeted his brothers when seeing them, they would have no need to stay in Egypt. However, by pretending not to know them and only later revealing himself as their brother, Yosef created an air of suspicion around the relationship he had with them. They had to stay in Egypt to let people know and to prove that they really are his brothers.
According to this we can explain what Gavriel reminded Yosef of. He didn’t remind Yosef that the brothers sold him, rather he reminded Yosef of the מדת הדין and the fact that the שעיבוד מצרים (slavery in Egypt) had to begin. This is what compelled Yosef pretend not to know them and accuse them of espionage. By doing this, he was ensuring that when he eventually identifies himself with them, they would have to remain in Egypt.
So possibly, on a surface level, Gavriel who represented the side of judgement was tied up with the story of Yosef. It was one that was predicated on din - judgement and led to the eventual enslavement of the Jewish people.